Some Useful Spherically Symmetric Profiles Used to Model Galaxies

Some useful spherically symmetric proﬁles used to model galaxies de Vaucouleurs’ R1/4 law for ellipticals and bulges

A good ﬁt to the light proﬁle of many ellipticals and bulges:

(constant such that Re encloses 1/2 light)

I = surface brightness R = projected radius

Re = effective radius enclosing 1/2 of light M87 Sérsic proﬁle

Generalization of de Vaucouleurs’ R1/4 law

n=Sérsic index (=4 for R1/4)

More luminous galaxies tend to have larger n

n=1 gives exponential proﬁle

which is a good approximation to many spirals 1990ApJ...356..359H 1990ApJ...356..359H

Hernquist proﬁle

Sky projection is close to R1/4 law, but 3D mass distribution is de Vaucouleurs analytically tractable (unlike R1/4): Hernquist

M = total mass a = scale radius

Convenient for theoretical models of ellipticals and bulges

Hernquist 90 Cosmological N-body simulations and dark matter halos The standard Λ Cold Dark Matter cosmology

Ωdm

Ωb

ΩΛ

Combined with other astronomical ‣ spectrum of initial density ﬂuctuations measurements, the cosmic microwave ‣ what the Universe is made out of background tells us: ‣ how old it is and how it has expanded in time Initial conditions for cosmological simulations

Microwave sky Simulation ICs 1x 4x

64x 16x

Visualization credit: Hahn

Density field in a 100 Mpc/h box with two initial levels of Example of an N-body simulation of a deeply nested re- refinement generated with MUSIC. gion of 6 initial levels generated with MUSIC and evolved Gaussian random field filtered with transfer function to with Gadget-2 to achieve an effective resolution of 81923 model early Universe physics (photon-baryon interactions) with 11603 particles in the high-res region. Introduction

Description of the method MUSIC (MUlti-Scale Initial Conditions) generates cosmological initial conditions for a hierarchical set of nested regions.

A detailed description of the method can be found in the code paper Hahn&Abel (2011), http://arxiv.org/abs/1103.6031. We kindly refer the reader to that paper for all technical aspects as well as performance and validation of the code.

Cookbook for setting up a zoom simulation with MUSIC The procedure for setting up a zoom simulation follows typically the procedure of 4 steps, given below. Note that resolution levels in MUSIC are speciﬁed by their linear power-2 exponent, i.e. a resolution of 1283 cells or particles

corresponds to level 7 (log2 128=7). We use the term “lower” for levels synonymously with “coarser” and “higher” with “ﬁner”.

Run a unigrid dark matter-only pre-ﬂight simulation

In order to set up unigrid initial conditions with MUSIC, select first the desired resolution for this pre-flight simulation. 3 Assume we want to run a 128 simulation, the coarse grid level has to be set to log2 128=7. Since we want to run a unigrid simulation, both levelmin and levelmax in section [setup] should be set to 7. Also the coarse grid seed needs to be chosen now and must not be changed afterwards. To do this, we set seed[7] in section [random] to the desired random seed. This seed determines the large scale structure and we will only add subgrid noise when performing refinement later. Now, set the box size, starting redshift, all the cosmological parameters and the input transfer function in the respective sections. Finally, select the output plugin in section [output] for the code with which you wish to perform this pre-flight simulation. Finally run MUSIC with the configuration file that contains all your settings and start your simulation. Note that you also have to explicitly specify a redshift at which to generate the initial conditions.

MUSIC - User's Manual 3 What is smoothed particle hydrodynamics?

DIFFERENT METHODS TO DISCRETIZE A FLUID What is smoothed particle hydrodynamics?

EulerianDIFFERENT METHODS TO DISCRETIZE ALagrangian FLUID discretize space Eulerian discretize massLagrangian representation on a mesh representation by fluid elements (volume elements)discretize space (particles) discretize mass representation onN a-body mesh simulations representation by fluid elements (volume elements) (particles) • Discretize mass with N particles

‣ in cosmology, usually tree or particle- mesh methods to solve Poisson’s equation principle advantage: principle advantage: high accuracy (shock capturing), low resolutions adjusts Naturally adaptive in cosmology numerical viscosityprinciple advantage:• automatically to theprinciple flow advantage: high accuracy (shock capturing), low resolutions adjusts numerical viscosity automatically to the flow collapse

collapse History of N-body simulations

Collisionless (cosmological)

Moore’s law

Collisional (e.g., globulars)

Dehnen & Read 11 Gravity ampliﬁes primordial ﬂuctuations, forms structures

Simplest: dark matter only

~100 Mpc

Visualization credit: Kravtsov

Density peaks (dark matter halos) are the sites of galaxy formation Millennium simulation (z=0 ﬂy through)

1010 particles, 500 h-1 Mpc Springel+05

NATURE|Vol 435|2 June 2005 ARTICLES

based on this formula may contain large errors15. We return below to black holes with a mass a billion times that of the Sun. At redshift the important question of the abundance of quasars at early times. z < 6, their co-moving space density is estimated to be To track the formation of galaxies and quasars in the simulation, ,(2.2 ^ 0.73) 1029h 3 Mpc23 (ref. 25). Whether such extremely we implement a semi-analytic model to follow gas, star and super- rare objects can£ form at all in a LCDM cosmology is unknown. massive black-hole processes within the merger history trees of dark A volume the size of the Millennium Simulation should contain, matter haloes and their substructures (see Supplementary Infor- on average, just under one quasar at the above space density. Just mation). The trees contain a total of about 800 million nodes, each what sort of object should be associated with these ‘first quasars’ is, corresponding to a dark matter subhalo and its associated galaxies. however, a matter of debate. In the local Universe, it appears that This methodology allows us to test, during postprocessing, many every bright galaxy hosts a supermassive black hole and there is a different phenomenological treatments of gas cooling, star for- remarkably good correlation between the mass of the central black mation, AGN growth, feedback, chemical enrichment and so on. hole and the stellar mass or velocity dispersion of the bulge of the Here, we use an update of models described in refs 16 and 17, which host galaxy26. It would therefore seem natural to assume that, at any are similar in spirit to previous semi-analytic models18–23;the epoch, the brightest quasars are always hosted by the largest galaxies. modelling assumptions and parameters are adjusted by trial and In our simulation, ‘large galaxies’ can be identified in various ways, error to fit the observed properties of low-redshift galaxies, primarily for example, according to their dark matter halo mass, stellar mass or their joint luminosity–colour distribution and their distributions of instantaneous star-formation rate. We have identified the ten ‘largest’ morphology, gas content and central black-hole mass. Our use of a objects defined in these three ways at redshift z 6.2. It turns out high-resolution simulation, particularly our ability to track the that these criteria all select essentially the same¼ objects: the eight evolution of dark matter substructures, removes much of the largest galaxies by halo mass are identical to the eight largest galaxies uncertainty of the more traditional semi-analytic approaches based by stellar mass; only the ranking differs. Somewhat larger differences on Monte Carlo realizations of merger trees. Our technique provides are present when galaxies are selected by star-formation rate, but accurate positions and peculiar velocities for all the model galaxies. It the four first-ranked galaxies are still among the eight identified also enables us to follow the evolutionary history of individual according to the other two criteria. objects and thus to investigate the relationship between populations In Fig. 3, we illustrate the environment of a ‘first quasar’ candidate seen at different epochs. It is the ability to establish such evolutionary in our simulation at z 6.2. The object lies on one of the most connections that makes this kind of modelling so powerful for prominent dark matter¼ filaments and is surrounded by a large interpreting observational data. number of other, much fainter galaxies. It has a stellar mass of 10 21 6.8 10 h M (, the largest in the entire simulation at z 6.2, a £ 12 21 ¼ The fate of the first quasars dark matter virial mass of 3.9 10 h M (, and a star-formation 21 £ Quasars are among the most luminous objects in the Universe and rate of 235M ( yr . In the local Universe, central black-hole masses can be detected at huge cosmological distances. Their luminosity is are typically ,1/1,000 of the bulge stellar mass27, but in the model we thought to be powered by accretion onto a central, supermassive test here these massive early galaxies have black-hole masses in the 8 9 black hole. Bright quasars have now been discovered as far back range 10 –10 M (, significantly larger than low-redshift galaxies of as redshift z 6.43 (ref. 24), and are believed to harbour central similar stellar mass. To attain the observed luminosities, they must Dark matter¼ halo mass functionconvert infalling mass to radiated energy with a somewhat higher efficiency than the ,0.1c 2 expected for accretion onto a non- spinning black hole (where c is the speed of light in vacuum). #Within halos our simulationper volume we can readilyper addressmass fundamental ques- tions such as: Where are the descendants of the early quasars today? Whatinterval were their progenitors? By tracking the merging history trees of the host haloes, we find that all our quasar candidates end up today as central galaxies in rich clusters. For example, the object depicted in Fig. 3 lies, today, at the centre of the ninth most massive cluster in the Dimensionless when15 expressed21 volume, of mass M 1.46 10 h M (.Thecandidatewith thein largestterms virial of ‘multiplicity mass¼ at£z 6.2 function’ (which has stellar mass 10 21 ¼ 12 21 4.7 10 h M (,virialmass4.85 10 h M (, and star-for- £ 21 £ mation rate 218M ( yr ) ends up in the second most massive cluster, 15 21 of mass 3.39 10 h M (. Following the merging tree backwards in time,Massive we can£ trace halos our quasar >M candidate* back to redshift z 16.7, 10 21 ¼ when its host halo had a mass of only 1.8 10 h M (. At this epoch,exponentially it is one of just 18 suppressed objects that we identify £(e.g., as collapsed systems with $20 particles. These results confirm the view that rich galaxy clustersgalaxy are ratherclusters special today) places. Not only are they the largest virialized structures today, they also lie in the regions where the first structures developed at high redshift. Thus, the best place to searchMW for halo the oldest ≈10 stars12 in M thesun Universe or for the descendants of the first supermassive black holes is at the centres of present-day rich Figure 2 | Differential halo number density as a function of mass and galaxy clusters. epoch. The function n(M, z) gives the co-moving number density of haloes less massive than M. We plot it as the halo multiplicity function ThePress-Schechter clustering evolution of darkis analytic matter and galaxies 2 21 M r dn/dM (symbols with 1-j error bars), where r is the mean density of The combination of a large-volume, high-resolution N-body simu- the Universe. Groups of particles were found using a friends-of-friends lationderivation; with realistic modellingbetter offits galaxies given enables by us to make precise algorithm6 with linking length equal to 0.2 of the mean particle separation. The fraction of mass bound to haloes of more than 20 particles (vertical theoreticalSeth-Tormen predictions forfunction the clustering of galaxies as a function of dotted line) grows from 6.42 1024 at z 10.07 to 0.496 at z 0. Solid redshift and intrinsic galaxy properties. These can be compared lines are predictions from an analytic£ fitting¼ function proposed¼ in previous directly with existing and planned surveys. The two-point correlation work11, and the dashed blue lines give the Press–Schechter model14 at function of our model galaxies at redshift z 0 is plotted in Fig. 4 z 10.07 and z 0. and is compared with a recent measurement¼ from the 2dFGRS ¼ ¼ Springel+05 631 © 2005 Nature Publishing Group Dark matter halo mass function

Press-Schechter analytic theory Seth-Tormen ﬁt 1996ApJ...462..563N 1996ApJ...462..563N 1996ApJ...462..563N 1996ApJ...462..563N

(Nearly) universal dark matter halo proﬁle

NFW proﬁle

ﬁts halos of all masses in N- body sims

Navarro, Frenk, White 96 1996ApJ...462..563N Concentration higher mass lower mass L ower-mass halosare more concentrated Navarro, Frenk, White 96 Concentration correlates with halo mass

Dark matter halo proﬁles form an (approximately) two-

parameter family (M200 and z)

[Historical note: original NFW paper focused on z=0 so said ‘one- parameter’ family]

Bullock+01 NFW496 proﬁle is a NAVARRO, generic FRENK, & WHITEoutcome of CDM Vol. 490 models

In CDM simulations, NFW emerges independent of cosmological parameters (e.g.,

Ωm≣Ω0) and power spectrum of initial conditions (spectral index n)

Still not fully understood

FIG. 2.ÈDensity proÐles of one of the most massive halos and one of the least massive halos in each series. In each panel, the low-mass system is represented by the leftmost curve. In the SCDM and CDM" models, radii are given in kiloparsecs (scale at top), and densities are in units of 1010 M_ kpc~3. In all other panels, the units are arbitrary. The density parameter,) , and the value of the spectral index, n, are given in each panel. The solid lines are Ðts to the density proÐles usingeq. (1). The arrows indicate the value of0 the gravitational softening. The virial radius of each system is in all cases 2 orders of magnitude larger than the gravitational softening. NFW97

P3M simulation, together with extra high frequency waves carries out simulations in physical, rather than comoving, added to Ðll out the power spectrum between the Nyquist coordinates and uses individual time steps for each particle. frequencies of the old and new particle grids. The regions The minimum time-step depends on the maximum density beyond the ““ high-resolution ÏÏ box are coarsely sampled resolved in each case, but it was typically 10~5H~1. with a few thousand particles of radially increasing mass in As discussed inNavarro et al. (1996), numerically0 con- order to account for the large-scale tidal Ðelds present in the vergent results require that the initial redshift of each run, original simulation. z , should be high enough that all resolved scales in the This procedure ensures the formation of a clump similar initialinit box are still in the linear regime. In order to satisfy in all respects to the one selected in the P3M run, except for this condition, we chosez so that the median initial dis- the improved numerical resolution. The size of the high- placement of particles ininit the high-resolution box was resolution box scales naturally with the total mass of each always less than the mean interparticle separation. Prob- system, and as a result all resimulated halos have about the lems with this procedure may arise ifz is so high that the same number of particles within the virial radius at z \ 0, gravitational softening (which is keptinit Ðxed in physical typically between 5000 and 10,000. The extensive tests pre- coordinates) becomes signiÐcantly larger that the mean sented inNavarro et al. (1996) indicate that this number of initial interparticle separation. We found this to be a particles is adequate to resolve the structure of a halo over problem only for the smallest masses,M M , in the [ * approximately two decades in radius. We therefore choose n \ 0,) \ 0.1 model. In this case, the initial redshift pre- 0 the gravitational softening,hg, to be 1% of the virial radius scribed by the median displacement condition is z [ 700, in all cases. (This is the scale length of a spline softening; see and the gravitational softening is then a signiÐcantinit fraction Navarro& White 1993 for a deÐnition.) The tree code of the initial box. This can a†ect the collapse of the earliest Aquarius (dark matter only) simulations

Springel+08

12 9 6 Mh~10 Msun (zoomed in) halos, ultra-high res. (up to 10 particles within Rvir) 1704 V. Springel et al.

2000; Fukushige & Makino 2001; Klypin et al. 2001; Jing & Suto that the enclosed mass for the Einasto profile is 2002; Fukushige & Makino 2003; Power et al. 2003; Fukushige, 3 α 4πr 2ρ 2 3 ln α 2 ln 8 3 2 r Kawai & Makino 2004; Navarro et al. 2004; Diemand et al. 2005; M(r) − − exp + − γ , , = α α α α r 2 Stoehr 2006; Knollmann, Power & Knebe 2008). ! " $ ! − " % It has often been claimed that the inner cusps of haloes and sub- (16) haloes may have slopes less than 1, with some studies even propos- where γ (a,x) is the lower incomplete gamma function. For a value − ing an asymptotic slope of 1.5 (Moore et al. 1999b; Fukushige of α 0.18 the radius where the maximum circular velocity peaks − = & Makino 2001). For main haloes this proposition has been ruled is given by rmax 2.189 r 2,andthemaximumcircularvelocityis = − 2 2 out in recent years by newer generations of simulations. Neverthe- related to the parameters of the profile by Vmax 11.19Gr 2 ρ 2. less, the idea that the asymptotic slope is typically steeper than 1 No published simulation to date has had enough= dynamic− range− − (e.g. 1.2) is still widespread and has been reiterated in recent to measure the logarithmic slope of the density profile in the region ∼− papers, even though this is clearly inconsistent with e.g. Fig. 4 or where the Einasto model would predict it to be shallower than 1, the numerical data in Navarro et al. (2004). so only indirect arguments could be advanced for this behaviour− With respect to the density profiles of subhaloes, the situation is (Navarro et al. 2004). This situation has changed with the Aquarius even more unclear. So far few studies have examined this question Project, as can be seen from Fig. 4, and in Navarro et al. (2008) directly. Stoehr (2006) found that the circular velocity curves of we provide a detailed analysis of this question. In the following, subhaloes are best fitted by a parabolic function relating log V to we focus on the density profiles of dark matter subhaloes,where log r, implying that the density profiles become shallower in the the available particle number is, of course, much smaller. Our best centre than NFW. On the other hand, Diemand et al. (2008) recently resolved subhaloes in the Aq-A-1 simulation contain more than 10 Downloaded from argued that subhaloes have steep cusps with a mean asymptotic slope million particles, allowing a relatively precise characterization of of 1.2. their density profiles. Until recently, such particle numbers repre- − We want to emphasize from the outset that the nature of halo and sented the state of the art for simulations of main haloes. subhalo density profiles, becoming gradually and monotonically In Fig. 22, we show spherically averaged density profiles for nine shallower towards the centre, makes it easy to arrive at the wrong subhaloes within the Aq-A halo. For each we compare up to five http://mnras.oxfordjournals.org/ conclusion for the structure of the inner cusp. Almost all numerical different resolutions, covering a factor of 1835 in particle mass. simulations to date have been able to produce demonstrably con- The density profiles line up quite well outside∼ their individual res- verged results for the density profile only in regions where the local olution limits, as predicted by the convergence criterion of Power slope is significantly steeper than 1. They have also all shown that et al. (2003) in the form given in equation (3). Individual profiles − the slope at the innermost measured point is significantly shallower in the panels are plotted as thick solid lines at radii where conver- than at radii a factor of a few further out. Thus, although no slope as gence is expected according to this criterion, but they are extended shallow as 1 has been found, there is also no convincing evidence inwards as thin lines to twice the gravitational softening length (the − that the values measured are close to the asymptotic value, if one gravitational force is exactly Newtonian outside the radii marked by exists. Most claims of steep inner cusp slopes are simply based on vertical dashed lines). These density profiles are based on particles the assertion that the slope measured at the innermost resolved point that are gravitationally bound to the subhaloes, but for comparison by guest on September 30, 2014 continues all the way to the centre. we also show a profile for each subhalo that includes all the mass Navarro et al. (2004) argued that the local logarithmic slope of (i.e. including unbound particles; thick dashed lines). It is clear halo profiles changes smoothly with radius and is poorly fitted by that the background density dominates beyond the ‘edge’ of each models like those of NFW or Moore that tend to an asymptotic subhalo. It is therefore important that this region is excluded when value on small scales. They showed that in their simulation data the fitting analytic model density profiles to the subhaloes. radial change of the local logarithmic slope can be well described In making such fits, we restrict ourselves to the radial range by a power law in radius, of the form between the convergence radius (equation 3) and the largest radius where the density of bound mass exceeds 80 per cent of the total α d log ρ r mass density. The density profiles themselves are measured in a set 2 , (14) Einasto profile d log r = − r 2 is better fit at small ofradius radial shells spaced equally in log r. To define the best fit, we ! − " Diversity and similarity of simulated CDM haloes 27 minimize the sum of the squared differences in the log between which corresponds to a density profile measurement and model, i.e. we characterize the goodness of fit by aquantity α 2 r 1 ρ(r) ρ 2 exp 1 . (15) Q2 [ln ρ ln ρmodel(r )]2, (17) = − − α r 2 − = N i − i # $! − " %& bins i ' Here ρ 2 and r 2 are the density and radius at the point where the where the sum extends over all bins i. We then minimize Q with local slope− is −2. This profile was first used by Einasto (1965) to respect to the parameters of the model profile. We have included describe the stellar− halo of the Milky Way, so we refer to it as the such fits as thin solid lines in Fig. 22, based on the Einasto profile, Einasto profile. The introduction of a shape parameter, α may be allowing the third parameter α to vary as well. The resulting values expected, of course, to provide improved fits, but we note that fixing of α and the maximum circular velocities of the subhaloes, as well α 0.16 gives a two-parameter function which still fits mean halo as their mass and distance to the main halo’s centre are shown as profiles∼ much better than the NFW form over a wide range of halo labels in the individual panels. masses (i.e. with maximum residuals of a few per cent rather than It is clear from Fig. 22 that the Einasto profile provides a good 10 per cent; Gao et al. 2007). Further evidence for a profile where description of subhalo radial density profiles, but due to the large local slope changes gradually has been presented by Stoehr et al. dynamic range on the vertical axis combined with the narrow radial (2003); Graham et al. (2006); Stoehr (2006). For reference, we note range over which the density profile can be fitted, it is not clear in Downloaded from

C 2008 The Authors. Journal compilation C 2008 RAS, MNRAS 391, 1685–1711 Figure 3. Left-hand panel: spherically averaged density profiles of all level-2 Aquarius haloes. Density estimates have been⃝ multiplied by r2 in order to ⃝ emphasize details in the comparison. Radii have been scaled to r 2,theradiuswherethelogarithmicslopehasthe‘isothermal’value, 2. Thick lines show (7) (1) − − the profiles from rconv outwards; thin lines extend inwards to rconv. For comparison, we also show the NFW and M99 profiles, which are fixed in these scaled units. This scaling makes clear that the inner profiles curve inwards more gradually than NFW, and are substantially shallower than predicted by M99. The bottom panels show residuals from the best fits (i.e. with the radial scaling free) to the profiles using various fitting formulae (Section 3.2). Note that the Einasto http://mnras.oxfordjournals.org/ formula fits all profiles well, especially in the inner regions. The shape parameter, α,variessignificantlyfromhalotohalo,indicatingthattheprofilesarenot strictly self-similar: no simple physical rescaling can match one halo on to another. The NFW formula is also able to reproduce the inner profiles quite well, although the slight mismatch in profile shapes leads to deviations that increase inwards and are maximal at the innermost resolved point. The steeply cusped Moore profile gives the poorest fits. Right-hand panel: same as the left, but for the circular velocity profiles, scaled to match the peak of each profile. This cumulative measure removes the bumps and wiggles induced by substructures and confirms the lack of self-similarity apparent in the left-hand panel.

Note: Einasto is Sérsic with I replace by ρ and R (projected) replaced by r (3D) α fixed to a single value, residuals are smaller and have less radial Navarro+10 structure than those from either NFW or M99. by guest on September 29, 2014 We show this in Fig. 4, where we plot the minimum-Q(Qmin) values of the best Einasto fits for all six level-2 Aquarius haloes, as a function of the shape parameter α.Forgivenvalueofα, the remaining two free parameters of the Einasto formula are allowed to vary in order to minimize Q2. Different line types correspond to different numbers of bins used to construct the profile (from 20 to 50), chosen to span in all cases the same radial range, 0.01

C C ⃝ 2009 The Authors. Journal compilation ⃝ 2009 RAS, MNRAS 402, 21–34 The Aquarius Project 1697

Einasto proﬁle (a ﬁt to our measurements yields a shape parameter 100 α 0.678 and scale radius r 2 199 kpc 0.81 r200). It is thus tempting= to conjecture that this− behaviour= continues= to (arbitrarily) small subhalo masses. If true, an interesting corollary is that there 10-1 must be a smooth dark matter component which dominates the inner regions of haloes. Only the outer parts may have a substantial mass fraction in lumps (see also Fig. 7). This contrasts with previous 10-2 speculations (Calcaneo-Rold´ an´ & Moore 2000; Moore et al. 2001)

that all the mass of a halo may be bound in subhaloes. loc sub f Further light on this question is shed by Fig. 12, where we show -3 Aq-A-2 10 the local mass fraction in subhaloes as a function of radius. In the Aq-B-2 top panel, we compare results for our six different haloes, with the Aq-C-2 radial coordinate normalized by r50. While there is some scatter Aq-D-2 -4 between the different haloes, the general behaviour is rather similar 10 Aq-E-2 and shows a rapid decline of the local mass fraction in substructures Aq-F-2 towards the inner parts of each halo. The mean of the six simulations (thick red line) is well ﬁtted by a gently curving power law. It can 10-5 be parametrized by 0.01 0.10 1.00 r / r50 Downloaded from 2 fsub exp γ β ln(r/r50) 0.5 α ln (r/r50) , (11) = + + 100 with parameters! α 0.36, β 0.87 and γ" 1.31. This ﬁt is shown in the upper= − two panels= of Fig. 12 as= a thin− black line. 10-1 The middle panel is the same measurement, but for all the different resolution simulations of the Aq-A halo, while the bottom panel -2 http://mnras.oxfordjournals.org/ is the corresponding cumulative plot. These two panels give an 10 loc sub impression of how well numerical convergence is achieved for this f quantity. 10-3 Aq-A-1 An interesting implication from Fig. 12 is an estimate of the frac- Aq-A-2 tion of the mass in substructures near the solar circle (marked by a 10-4 Aq-A-3 vertical dashed line). At r 8 kpc, the expected local mass fraction Aq-A-4 = 3 Aq-A-5 in substructure has dropped well below 10− .Thismeasurementap- 10-5 pears converged, and accounting for unresolved substructure does Dark matter1 substructure10 100 3 r [ kpc ] not raise the fraction above 10− (compare Fig. 7). The dark mat- Cumulative mass fraction in sub-halos by guest on September 30, 2014 ter distribution through which the Earth moves should therefore be 0 mostly smooth, with only a very small contribution from gravita- 10 sub-halos get disrupted by tidal tionally bound subhaloes. forces as they sink into parent halos -1 10 owing to dynamical friction

4SUBHALOESINSIDESUBHALOES -2 ) 10 See Mike’s review. r

In our simulations, we ﬁnd several levels of substructure within (< How many sub-halos in MW?

cumul sub -3 Extrapolations substructure. Fig. 13 illustrates this by showing individually six of f 10 the largest Aq-A-1 subhaloes in enlarged frames. Clearly, all of these Aq-A-1 subhaloes have embedded substructures. Sometimes these second- Aq-A-2 Important for dark matter annihilation predictions, 2-body 10-4 Aq-A-3 so sensitive to boost factor generation subhaloes contain a further (third) level of substructure Aq-A-4 and, in a few cases, we even find a fourth generation of subhaloes Aq-A-5 10-5 embedded within these. An example is given in the bottom row of 1 10 100 Fig. 13, which zooms recursively on regions of the subhalo labelled r [ kpc ] ‘f’ in the top left-hand panel. As shown in the bottom left-hand panel, subhalo ‘f’ has several components, each of which has identifiable Figure 12. The mass fraction≈10% in halo subhaloes mass in as sub-halos a function of radius. In the top panel, we show results for the local mass fraction in substructures for subcomponents; we are able to identify up to four levels of this Springel+08 our six different haloes, as a function of radius normalized by r50.The hierarchy of substructure in this system. We note that the hierarchy thick solid line shows the average of all the runs. In the middle panel, we of nested structures is established directly by the recursive nature of consider the same quantity for the different resolution simulations of the Aq- the SUBFIND algorithm; at each level, a given substructure and its A halo, while in the bottom panel we show the corresponding cumulative parent structure are surrounded by a common outer density contour substructure fractions in the Aq-A halo. The solid line in the two upper panels that separates them from the next level in the hierarchy. is an empirical fit with a slowly running power-law index. The vertical dotted It is important to quantify in detail the hierarchical nature of lines at 8 kpc in the middle and bottom panels mark the position of the solar substructure, since this may have a number of consequences re- circle; here the expected local mass fraction in subhaloes has dropped well 3 garding indirect and direct dark matter search strategies. Recently, below 10− .Theouterverticaldottedlinesmarkr50 for the Aq-A halo. Shaw et al. (2007) suggested that the (sub)substructure distribution in subhaloes might be a scaled version of the substructure distribution in main haloes. This claim has been echoed by Diemand et al. (2008), who report roughly equal numbers of substructures

C C ⃝ 2008 The Authors. Journal compilation ⃝ 2008 RAS, MNRAS 391, 1685–1711 The missing satellites “problem" (of DM-only sims)

Klypin+99

FIG. 4.ÈProperties of satellite systems within 200 h~1 kpc from"Cosmological the FmodelsIG. 5.ÈSame thus predict as in Fig. that 4, a but halo for the satellites size of within our Galaxy 400 h ~1shouldkpc from have the about 50 dark host halo. T op: The three-dimensional rms velocity dispersion of satellitesmatter satellitescenter with ofcircular a host velocity halo. In greater the bottom than 20 panel km s we-1 and also mass show greater the cumulative than 3×108 Msun within vs. the maximum circular velocity of the central halo. Solid anda 570 open kpc radius.velocity This functionnumber foris signiﬁcantly the Ðeld halos higher (halos than outside the approximately of 400 h~1 kpc dozen spheres satellites actually circles denote "CDM and CDM halos, respectively. The solid line is the around seven massiveobserved halos), arbitrarily around our scaled Galaxy. up by” a factor of 75. The line of equal satellite rms velocity dispersion and the circular velocity of the di†erence at large circular velocitiesV [ 50 km s~1 is not statistically circ host halo. Middle: The number of satellites with circular velocityNo larger longer a realsigni "problem":Ðcant. Comparison more faint betweendwarfs now these detected, two curves baryonic indicates effects that the(photoionization, veloc- stellar than 10 km s~1 vs. circular velocity of the host halo. The solid line showsfeedback) a canity suppress functions dwarf of isolated galaxy and formation satellite in halos low-mass are very halos similar. and affect As for their the inner proﬁles rough approximation presented in the legend. Bottom: The cumulative satellites within the central 200 h~1 kpc (Fig. 4), the number of satellites in circular VDF of satellites. Solid triangles show average VDF of MW and the models and in the Local Group agrees reasonably well for massive Andromeda satellites. Open circles present results for the CDM simula- satellites withV [ 50 km s~1 but disagrees by a factor of 10 for low- circ tion, while the solid curve represents the average VDF of satellites in the mass satellites withV \ 10È30 km s~1. circ "CDM simulation for halos shown in the upper panels. To indicate the statistics, the scale on the right y-axis shows the total number of satellite halos in the "CDM simulation. Note that while the numbers of massive satellites ([50 km s~1) agree reasonably well with the observed number of satellites in the Local Group, models predict about 5 times more lower mass satellites withV \ 10È30 km s~1. circ

TABLE 3 SATELLITES IN "CDM MODEL INSIDE R \ 200/400 h~1 kpc FROM CENTRAL HALO

Halo V Halo Mass V V circ rms rotation (km s~1)(h~1M_) Number of Satellites Fraction of Mass in Satellites (km s~1) (km s~1) 140.5 ...... 2.93 ] 1011 9/15 0.053/0.112 99.4/94.4 28.6/15.0 278.2 ...... 3.90 ] 1012 39/94 0.041/0.049 334.9/287.6 29.8/11.8 205.2 ...... 1.22 ] 1012 27/44 0.025/0.051 191.7/168.0 20.0/11.3 175.2 ...... 6.26 ] 1011 5/10 0.105/0.135 129.1/120.5 41.5/45.2 259.5 ...... 2.74 ] 1012 24/52 0.017/0.029 305.0/257.3 97.1/16.8 302.3 ...... 5.12 ] 1012 37/105 0.055/0.112 394.6/331.6 39.4/15.7 198.9 ...... 1.33 ] 1012 24/58 0.048/0.049 206.1/169.3 17.7/12.1 169.8 ...... 7.91 ] 1011 17/26 0.053/0.067 162.8/156.0 9.3/5.0 Populating halos with galaxies

-1 z=0.1 z=1

Three approaches to connect N- -2 ] body simulations to galaxies: -1 -3 dex -3 -4

-5 ) [Mpc ‣ star -6 empirical halo-based models m -1 z=2 z=4 10 ‣ semi-analytic models -2 -3 ‣ hydro simulations log dN/(dlog -4

-5

-6 Now comparably successful at 8 9 10 11 12 8 9 10 11 12 log mstar [Msun] Frontier in galaxy formation is now to replace model parameters with

producing galaxy populations Figure 4: explicit, physical models for all the important processes shaping galaxies Galaxy stellar mass function at redshifts z ∼ 0–4. In the z =0.1, z =1,andz =2panels,blacksquare symbols show a double-Schechter fit to a compilation of observational estimates. Observations included broadly consistent with in the fit are: z =0.1–Baldry,Glazebrook&Driver(2008),Moustakasetal.(2013); z =1andz =2 panels – Tomczak et al. (2014), Muzzin et al. (2013). The fits shown at z =1andz =2areinterpolated to these redshifts from adjacent redshift bins in the original published results. The formal quoted 1σ observations but different degrees errors on the estimates shown in these three panels are comparable to the symbol size, and are not shown for clarity (the actual uncertainties are much larger,butaredifficult to estimate accurately). In the z =0.1panel,theestimatesofBernardietal.(2013)arealsoshown(opengraycircles).Inthe z =4panelweshowestimatesfromDuncanetal.(2014,triangles), Caputi et al. (2011, crosses), of predictive power and important Marchesini et al. (2010, circles, for z =3–4),andMuzzinetal.(2013,pentagons,z =3–4).Solidcolored lines show predictions from semi-analytic models: SAGE (Croton et al. in prep, dark blue), Y. Lu SAM (Lu et al. 2013, magenta), GALFORM (Gonzalez-Perez et al. 2014, green), the Santa Cruz SAM (Porter et al. 2014, purple), and the MPA Millennium SAM (Henriques etal.2013).Thedottedlightblueline differences in detailed predictions shows the Henriques et al. (2013) SAM with observational errors convolved (see text). Colored dashed lines show predictions from numerical hydrodynamic simulations: EAGLE simulations (Schaye et al. 2014, dark red), ezw simulations of Davéand collaborators (Davéet al. 2013, bright red) and the (esp., gas quantities) Illustris simulations (Vogelsberger et al. 2014b, orange). Somerville & Davé (2015) review brush sense the model predictions are generally encouraging. A very general prediction of ΛCDM-based models is that galaxies built up their stellar mass gradually over time, which is supported by observations. All models predict efficient early star formation (z > 4) in low mass halos, and steep stellar mass and rest-UV luminosity functions at these early∼ epochs, in agreement with observations. Models including AGN feedback or heuristic quenching predict that massive galaxies formed earlier and more rapidly than lower mass galaxies. This is again in qualitative agreement with observations. Most models even demonstrate

34 Somerville & Davé Abundance matching 906 MOSTER ET AL. Vol. 710 line of sight: halo mass function Assume monotonic relationship rξ r ∞ 2 2 ∞ ( ) (offset) wp(rp) 2 dr ξ r + rp 2 dr , || 2 2 = 0 || = rp r r between stellar mass and dark ! "# $ ! − p # (4) matter halo mass, i.e. where the comoving distance (r)hasbeendecomposedinto components parallel (r )andperpendicular(rp)totheline || galaxy stellar mass of sight. The integration is truncated at 45 Mpc. Due to the finite size of the simulation box (Lbox 100 Mpc), the function model correlation function is not reliable= beyond scales of in given volume, highest stellar r 0.1Lbox 10 Mpc. ∼In order to∼ fit the model to the observations, we use Powell’s mass is assigned to most directions set method in multidimensions (e.g., Press et al. 1992) to find the values of M1,(m/M)0, β,andγ that minimize either

massive dark halo, … 2 2 2 χ (Φ) χr χr (Φ) = = NΦ (mass function ﬁt) or

Comparison of stellar mass 2 2 2 2 2 χ (Φ) χ (wp) χr χr (Φ)+χr (wp) + and dark matter halo mass = = NΦ Nr Nm Figure 1. Comparison between the halo mass function offset by a factor of 0.05 (mass function and projected CF ﬁt) with N and N the number (dashed line), the observed galaxy mass function (symbols), our model without Φ r functions yields Ṃ/Mhalo scatter (solid line), and our model including scatter (dotted line).We see that of data points for the SMF and projected CFs, respectively, and the halo and the galaxy mass functions are different shapes, implying that the Nm the number of mass bins for the projected CFs. 2 2 stellar-to-halo mass ratio m/M is not constant. Our four-parameter model for In this context, χ (Φ)andχ (wp)aredeﬁnedasfollows: relationship the halo mass dependent stellar-to-halo mass ratio is in very good agreement

with the observations (both including and neglecting scatter). NΦ 2 Φmod(mi ) Φobs(mi ) χ 2(Φ) − , 3.2. Constraining the Free Parameters Moster+10 = σ i 1 & Φobs(mi ) ' %= Having set up the model, we now need to constrain the four Nm Nr 2 2 wp,mod(rp,j ,mi ) wp,obs(rp,j ,mi ) free parameters M1,(m/M)0, β,andγ .Todothis,wepopulate χ (wp) − , = σ the halos in the simulation with galaxies. The stellar masses of i 1 j 1 & wp,obs(rp,j ,mi ) ' the galaxies depend on the mass of the halo and are derived %= %=

according to our prescription (Equation (2)). The positions with σΦobs and σwp,obs the errors for the SMF and projected CFs, of the galaxies are given by the halo positions in the N-body respectively. Note that for the simultaneous fit, by adding the 2 simulation. reduced χr ,wegivethesameweighttobothdatasets. Once the simulation box is filled with galaxies, it is straight- 3.3. Estimation of Parameter Errors forward to compute the SMF Φmod(m). As we want to fit this model mass function to the observed mass function Φobs(m) In order to obtain estimates of the errors on the parameters, by Panter et al. (2007), we choose the same stellar mass range we need their probability distribution prob(A I), where A is the 8.5 11.85 (10 M Ð10 M )andthesamebinsize.Theobserved parameter under consideration and I is the| given background ⊙ ⊙ SMF was derived using spectra from the Sloan Digital Sky information. The most likely value of A is then given by: Survey Data Release 3 (SDSS DR3); see Panter et al. (2004)for Abest max(prob(A I)). adescriptionofthemethod. As= we have to assume| that all our parameters are coupled, we Furthermore, it is possible to determine the stellar mass can only compute the probability for a given set of parameters. dependent clustering of galaxies. For this, we compute projected This probability is given by: galaxy CFs wp,mod(rp,mi )inseveralstellarmassbinswhichwe 2 choose to be the same as in the observed projected galaxy CFs of prob(M1, (m/M)0, β, γ I) exp( χ ). Li et al. (2006). These were derived using a sample of galaxies | ∝ − from the SDSS DR2 with stellar masses estimated from spectra In a system with four free parameters A, B, C,andD one can by Kauffmann et al. (2003). calculate the probability distribution of one parameter (e.g., A) We first calculate the real space CF ξ(r). In a simulation, this if the probability distribution for the set of parameters is known, can be done by simply counting pairs in distance bins: using marginalization:

dd(r ) ∞ ξ(r ) i 1, (3) prob(A I) prob(A, B I)dB i N (r ) | = | = p i − !−∞ ∞ prob(A, B, C, D I)dB dC dD . where dd(ri )isthenumberofpairscountedinadistancebin 2 2 3 = | and Np(ri ) 2πN r ∆ri /L ,whereN is the total number of !−∞ = i box galaxies in the box. The projected CF wp(rp)canbederived Once the probability distribution for a parameter is deter- by integrating the real space correlation function ξ(r)alongthe mined, one can assign errors based on the conﬁdence intervals. No. 2, 2010 STELLAR-TO-HALO MASS RELATIONSHIP 909

Abundance matching result at z=0

Stellar - dark matter halo mass relation

stellar 0.2 Ωb/Ωm feedback

AGN

halo feedback? M / ̣ M log log

Moster+10

log Mhalo (Msun) Figure 4. Derived relation between stellar mass and halo mass. The light shaded area shows the 1σ region while the dark and light shaded areas together show the 2σ region. The upper panel shows the SHM relation, whileRadio the lowerjet panel Figure 5. Correlations between the model parameters. The panels show contours Galactic showswind thein SHMM82 ratio.starburst of constant χ 2 (i.e., constant probability) for the ﬁt including constraints from the SMF only. The parameter pairs are indicated in each panel.

mainly the slope of the low mass end of the SMF, it is strongly related to the parameter α of the Schechter function. A small Table 2 Fitting Results for Stellar-to-halo Mass Relationship value of β corresponds to a high value of α. 2 2 If we change γ ,thismainlyimpactstheslopeofthemassive log M1 (m/M)0 βγχr (Φ) χr (wp) end of the SMF. For larger values of γ than for its best-fit value, Best fit 11.899 0.02817 1.068 0.611 1.42 4.21 the slope of the massive end becomes steeper. As γ affects σ + 0.026 0.00063 0.051 0.012 mainly the slope of the massive end of the SMF, it is not coupled σ − 0.024 0.00057 0.044 0.010 to a parameter of the Schechter function though it is related to the high-mass cutoff, assumed to be exponential in a Schechter Notes. Including scatter σm 0.15. All masses are in units of M . = ⊙ function. Figure 5 shows the contours of the two-dimensional probability distributions for the parameters pairs. We see a correlation halo occupation models, SAMs and satellite kinematics (Cooray 2006;vandenBoschetal.2007;Moreetal.2009b). between the parameters [M1, γ ]and[(m/M)0, γ ]andananti- Assuming a value of σ 0.15 dex and fitting the SMF only, correlation between [β, γ ], [β,M1], and [(m/M)0,M1]. There m we find the values given in= Table 2.Thesevaluesliewithinthe does not seem to be a correlation between [β, (m/M)0]. (2σ )errorbarsofthebest-fitvaluesthatweobtainedwithno 4.5. Introducing Scatter scatter. The largest change is on the value of γ ,whichcontrols the slope of the SHM relation at large halo masses. The SMF Up until now we have assumed that there is a one-to-one, and the projected CFs for the model including scatter are shown deterministic relationship between halo mass and stellar mass. in Figures 1 and 2,respectively,andshowverygoodagreement However, in nature, we expect that two halos of the same mass M with the observed data. may harbor galaxies with different stellar masses, since they can In Figure 6,wecompareourmodelwithoutscatterwiththe have different halo concentrations, spin parameters, and merger model including scatter. We have also included the relation histories. between halo mass and the average stellar mass. Especially For each halo of mass M,wenowassignastellarmassm at the massive end scatter can influence the slope of the SMF, drawn from a lognormal distribution with a mean value given since there are few massive galaxies. This has an impact on γ by our previous expression for m(M)(Equation(2)), with a and as all parameters are correlated scatter also affects the other 2 variance of σm.Weassumethatthevarianceisaconstantfor parameters. We thus see a difference between the model without all halo masses, which means that the percent deviation from scatter and the most likely stellar mass in the model with scatter m is the same for every galaxy. This is consistent with other in Figure 6. Galaxies are much smaller than halos

Can use abundance matching to connect galaxy to halo properties

At z=0,

Kravtsov 2013 Semi-analytic models (SAMs)

Assume galaxies form inside dark matter halos → merger trees

Use simple analytic models (‘recipes’) to determine galaxy properties ‣ gas cooling ‣ angular momentum of disk ‣ effects of feedback, …

Analytic prescriptions do not capture full complexity of galaxy formation, so parameters are tuned to match observations

Figure 3: Computationally inexpensive, so Visualization of representative predictions from a semi-analytic model. Symbol sizes represent the mass of the host dark matter halo; the x-axis is arbitrary. Symbolsconnectedbylinesrepresenthalomergers. can systematically explore Colors represent the mass of different galaxy components (red: hot gas; blue: cold gas; yellow: stars). Several different final host halo masses are shown as indicatedonthefigurepanels.Reproducedfrom parameter space Hirschmann et al. (2012a). Somerville & Davé (2015) review that can’t be simulated explicitly, and tuning these parameters to match a subset of observations, 2) experimenting with different sub-grid recipes toachievethebestmatchtoaset of observations. Even the recipes themselves are in many cases very similar to the ones that are commonly implemented in SAMs. Encouragingly, the two techniques have arrived at the same qualitative conclusions about galaxy formation andevolutionforallofthetopics that we will discuss in this article. For this reason, we structure this article largely in terms of the physical processes and general insights into how they shape galaxy formation, giving examples from both SAMs and numerical simulations.

Models of Galaxy Formation 13 Hydrodynamical simulations of galaxy formation Explicitly include baryons in the Gas simulation and attempt to explicitly capture all important physics Movie of Limited by computational power, FIRE zoom-in so still need to rely on ‘sub- of MW-like resolution models’ for processes galaxy occurring below resolution limit Stars (e.g., star formation)

Large-volume (~100 Mpc) hydro sims have typical resolution ~1 kpc; zoom-ins ~10 pc

Most detailed predictions, e.g. gas around galaxies (the circum- galactic and intergalactic media) The Latte cosmological simulation of a Milky Way analog

140 million particles total (mb=7,070 Msun, mDM=3.5×104 Msun)

u / g / r mock images of stellar light Wetzel+16 Latte Aitoff projection from the “solar" circle

Wetzel+16 Baryonic physics (including stellar feedback) resolves the missing satellites problem in Latte

Wetzel+16

INSIGHT REVIEW NATURE|Vol 440|27 April 2006

h Galaxy clustering 12h 11 Figure 1 | The galaxy distribution obtained from 10 h h spectroscopic redshift surveys and from mock 13 catalogues constructed from cosmological 9 h simulations. When populated with h The small slice at the top shows the 14 CfA2 ‘Great Wall’3, with the Coma cluster at the Sloan Great Wall galaxies, the cosmic centre. Drawn to the same scale is a small section 25,000 of the SDSS, in which an even larger ‘Sloan 100 0 –1 ) s +SAM Great Wall’ has been identified . This is one of 2dF n h h h 20,00 3 web of dark matter h 13 12 (km h the largest observed structures in the Universe, GRS h z 22 14 11 c h 10 Millennium 15 h containing over 10,000 galaxies and stretching 1.5 simulatio h 15,000 9 h 1.5 16 over more than 1.37 billion light years. The cone predicted by h Billion years 8 h h ion years 2 17 h on the left shows one-half of the 2dFGRS, which 3 2 n bill 1.0 e i 1.0 determined distances to more than 220,000 CfA2 Great Wall k tim cosmological N-body 10,000c kba galaxies in the southern sky out to a depth of Loo 0.5 2 billion light years. The SDSS has a similar

1 h 5,000 0.5 h simulations explains the 0 depth but a larger solid angle and currently includes over 650,000 observed redshifts in the northern sky. At the bottom and on the right,

observed clustering of 0

h 1 mock galaxy surveys constructed using semi- 0.20 5,0 analytic techniques to simulate the formation galaxies 0.05 00 and evolution of galaxies within the evolving 10

,000 0.1 23 dark matter distribution of the ‘Millennium’

h 5 2

0.10 h 5 z Redshift simulation are shown, selected with matching ft 8 h h 17 0.10 survey geometries and magnitude limits. 5 edshi h 0.1 R 9 z

h 16

15,00 22

h 10 h c 3 h

ion 15 z h → strong evidence in 11 h h 0 (k 0.0.5 h h 14 m s 0.20 12 13 2 0, –1 imulat 00 ) support of gravitational 0 ium s 2 5, 000

instability-induced large- Millenn

h 9

h scale structure as 10

11 12 h predicted by ΛCDM h

When expressed in units of the critical density required for Springel,a flat cos- Frenk,low estimates White of the 06 mean matter density Ωm are incompatible with the mic geometry, the mean density of dark matter is usually denoted by flat geometry predicted by inflation unless the Universe contains an Ωdm. Although a variety of dynamical tests have been used to constrain additional unclustered and dominant contribution to its energy density, Ωdm, in general such tests give ambiguous results because velocities are for example a cosmological constant Λ such that Ωm + ΩΛ ≈ 1. Two large- induced by the unseen dark matter and the relation of its distribution scale structure surveys carried out in the late 1980s, the APM (automated to that of the visible tracers of structure is uncertain. The notion of a photographic measuring) photographic survey23 and the QDOT redshift substantial bias in the galaxy distribution relative to that of dark matter survey of infrared galaxies24, showed that the power spectrum of the was introduced in the 1980s to account for the fact that different samples galaxy distribution, if it traces that of the mass on large scales, can be of galaxies or clusters are not directly tracing the underlying matter fitted by a simple CDM model only if the matter density is low, Ωm ≈ 0.3. distribution15–17. Defined simply as the ratio of the clustering strengths, This independent confirmation of the dynamical arguments led many the ‘bias function’ was also invoked to reconcile low dynamical estimates to adopt the now standard model of cosmology, ΛCDM. for the mass-to-light ratio of clusters with the high global value required It was therefore with a mixture of amazement and déjà vu that cos- in the theoretically preferred flat, Ωdm = 1 universe. But because massive mologists greeted the discovery in 1998 of an accelerated cosmic expan- clusters must contain approximately the universal mix of dark matter sion25,26. Two independent teams used distant type Ia supernovae to and baryons (ordinary matter), this uncertainty is neatly bypassed by perform a classical observational test. These ‘standard candles’ can be comparing the measured baryon fraction in clusters with the universal observed out to redshifts beyond 1. Those at z ≥ 0.5 are fainter than fraction under the assumption that the mean baryon density, Ωb, is the expected, apparently indicating that the cosmic expansion is currently value inferred from Big Bang nucleosynthesis18. Applied to the Coma speeding up. Within the standard Friedmann cosmology, there is only cluster, this simple argument gave Ωdm ≤ 0.3 where the inequality arises one agent that can produce an accelerating expansion: the cosmological because some or all of the dark matter could be baryonic18. This was constant first introduced by Einstein, or its possibly time- or space- the first determination of Ωdm < 1 that could not be explained away by dependent generalization, ‘dark energy’. The supernova evidence is 19 invoking bias. Subsequent measurements have confirmed the result consistent with ΩΛ ≈ 0.7, just the value required for the flat universe which also agrees with recent independent estimates based, for example, predicted by inflation. on the relatively slow evolution of the abundance of galaxy clusters20,21 or The other key prediction of inflation, a density fluctuation field con- on the detailed structure of fluctuations in the microwave background sistent with amplified quantum noise, received empirical support from radiation22. the discovery by the COsmic Background Explorer (COBE) satellite in The mean baryon density implied by matching Big Bang nucle- 1992 of small fluctuations in the temperature of the cosmic microwave osynthesis to the observed abundances of the light elements is background (CMB) radiation27. These reflect primordial density fluc- 2 only Ωbh ≈ 0.02, where h denotes the Hubble constant in units of tuations, modified by damping processes in the early Universe which 100 km s–1 Mpc–1. Dynamical estimates, although subject to bias uncer- depend on the matter and radiation content of the Universe. More recent 28–32 tainties, have long suggested that Ωm = Ωdm + Ωb ≈ 0.3, implying that the measurements of the CMB culminating with those by the WMAP dark matter cannot be baryonic. Plausibly it is made up of the hypotheti- (Wilkinson Microwave Anisotropy Probe) satellite22 have provided a cal elementary particles postulated in the 1980s, for example axions or striking confirmation of the inflationary CDM model: the measured the lowest mass supersymmetric partner of the known particles. Such temperature fluctuation spectrum is nearly scale-invariant on large 1138 © 2006 Nature Publishing Group

The DASI experiment (Pryke et al.,2002)gives of Bright Galaxies, the Zwicky catalog, the Lick catalog

+0.08 and later on the very deep Jagellonian field (Peebles, 1975; n =1.01−0.06 (9) Peebles and Groth, 1975; Peebles and Hauser, 1974). All this where the error bars are 68%confidencelimits.Thisresult work was done on the projected distribution of galaxies since comes from fitting the DASI data alone, making typical prior little or no redshift information was available. assumptions about such things as the Hubble constant. The The central discovery was that the two-point correlation recent WMAP data gives a value function describing the deviation of the galaxy distribution from homogeneity scales like a simple power law over a sub- n =0.99 0.04 (10) stantial range of distances. This result has stood firm through ± numerous analyses of diverse catalogs over the subsequent (Spergel et al.,2003). (Thislattervaluecomesfromthe decades. NATURE|Vol 435|2 June 2005 ARTICLES WMAP data alone, no other data is taken into account.) The amplitudes of the correlation functions calculated from

Other similar numbers come from Wang et al. (2002) and 39 the different catalogs were found to scale in accordancethan was wit possibleh in earlier work : almost five orders of magnitude in the Lyman-break galaxies observed at z < 3 (ref. 41). Signatures of Miller et al. (2002b). the nominal depth of the catalog. This was one of thewavenumber first di- k. the first two acoustic peaks are clearly visible at both redshifts, even It is perhaps appropriate to point out that this fit comes At present, the acoustic oscillations in the matter power spectrum though the density field of the z 3 galaxies is much more strongly rect proofs that the Universe is homogeneous. Beforeare that expected we to fall in the transition region between linear and biased with respect to the dark¼ matter (by a factor b 2.7, where 1/2 ¼ from data on scales bigger than the scale of significant galaxy knew about the isotropy of the galaxy distribution at differnonlinearent scales. In Fig. 6, we examine the matter power spectrum b [Pgal(k)/Pdm(k)] ) than at low redshift. Selecting galaxies by in our simulation in the region of the oscillations. Dividing by the their¼ star-formation rate rather than their stellar mass (above clustering and that it is a matter of belief that the primordial 40 21 depths and could only infer homogeneity by arguingsmooth that we power spectrum of a LCDM model with no baryons 10.6M ( yr for an equal space density at z 3) produces very power law continued in the same manner to smaller scales. In were not at the center of the Universe. highlights the baryonic features in the initial power spectrum of similar results. ¼ fact, more complex inflationary models predict a slowly vary- the simulation, although there is substantial scatter owing to the Our analysis demonstrates conclusively that baryon wiggles small number of large-scale modes. Because linear growth preserves should indeed be present in the galaxy distribution out to redshift ing exponent (spectral index) (see, e.g., Kosowsky and Turner the relative mode amplitudes, we can approximately correct for this z 3. This has been assumed, but not justified, in recent proposals to (1995)); this is in accordance with the WMAP data. The scatter by scaling the measured power in each bin by a multiplicative use¼ evolution of the large-scale galaxy distribution to constrain B. The correlation function: galaxies factor based on the initial difference between the actual bin power the nature of the dark energy. To establish whether the baryon scales which are relevant to the clustering of galaxies are just and the mean power expected in our LCDM model. This makes the oscillations can be measured in practice with the requisite accu- those scales where the effects of the recombination process 1. Definitions and scaling effects of nonlinear evolution on the baryon oscillations more clearly racy will require detailed modelling of the selection criteria of an visible. As Fig. 6 shows, nonlinear evolution not only accelerates actual survey, and a thorough understanding of the systematic on the fluctuation spectrum are the greatest. We believe we growth but also reduces the baryon oscillations: scales near peaks effects that will inevitably be present in real data. These issues can understand that process fully (Hu et al.,2001,1997)andso The definition of the correlation function used ingrow cosmol- slightly more slowly than scales near troughs. This is a only be properly addressed by means of specially designed mock consequence of the mode–mode coupling characteristic of nonlinear we have no hesitation in saying what are the consequences of ogy differs slightly from the definition used in other fields.growth.In spite of these effects, the first two ‘acoustic peaks’ (at having an initial n =1power spectrum. That, and the success cosmology we have a nonzero mean field (the meank density< 0.07 and k < 0.13h Mpc21, respectively) in the dark matter distribution do survive in distorted form until the present day (see of the N-body experiments, provide a good basis for the belief of the Universe) superposed on which are the fluctuationsFig. 6f). that that n 1 on galaxy clustering scales. Anyway, it is probable correspond to the galaxies and galaxy clusters. Since theAre Un thei- baryon wiggles also present in the galaxy distribution? ≈ Fig. 6 shows that the answer to this important question is “yes”. The that the Sunyaev-Zel’dovich effect (Sunyaev and Zel’dovich, verse is homogeneous on the largest scales, the correlationz 0 panels (Fig. 6f) shows the power spectrum for all model galaxies ¼ 1980) will dominate on the scales we are interested in so we tend to zero on these scales. brighter than M B 217 (M B is the magnitude in the optical blue waveband). On the¼ largest scales, the galaxy power spectrum has the may never see the recombination-damped primordial fluctua- On occasion, people have tried to use the standard definitionsame shape as that of the dark matter, but with slightly lower amplitude corresponding to an ‘antibias’ of 8%. Samples of brighter tions on such scales. and in doing so have come up with anomalous conclusions. galaxies show less antibias, while for the brightest galaxies, the bias We therefore have a classical initial value problem: the dif- The right definition is: In cosmology, the 2-pointbecomes galaxy slightly positive. Figure 6d, e also shows measurements of ficulty lies mainly in knowing what physics, subsequent to correlation function is defined as a measure of thethe excess power spectrum of luminous galaxies at redshifts z 0.98 and z 3.06. Galaxies at z 0.98 were selected to have a¼ magnitude recombination, our solution will need as input and knowing ¼ ¼ probability, relative to a Poisson distribution, of findingM B ,2two19 in the restframe, whereas galaxies at z 3.06 were 9 2¼1 how to compare the results of the consequent numerical sim- galaxies at the volume elements dV1 and dV2 separatedselected by to a have stellar mass larger than 5.83 10 h M (, corresponding to a space density of 8 1023h 3 Mpc£23, similar to that of ulations with observation. CMB measurements can also give vector distance r: £ us valuable clues for these later epochs in the evolution of the Correlations functions 2 universe. A good example is the discovery of significant large- dP12 = n [1 + ξ(r)]dV1dV2, (11) scale CMB polarization by the WMAP team (Kogut et al., 2003) that pushes the secondary re-ionization (formation of where n is the mean number density over the whole sample the first generation of stars) back to redshifts z 20. volume. When homogeneity5 and isotropy are assumed ξ(r) ≈ depends only onClustering the distance is mader = quantitativer .FromEq.(11),itis straightforward to derive the expression| | for the conditional using correlation functions V. MEASUREMENTS OF CLUSTERING probability that a galaxy lies at dV at distance r given that there is a galaxy at the origin of r. A. The discovery of power-law clustering dP = n[1 + ξ(r)]dV. (12) The pioneering work of Rubin and Limber has already been mentioned. These early authors were limited by the nature of Therefore, ξ(r) measures the clustering in excess (ξ(r) > 0) the catalogs that existed at the time and the means to analyze or in defect (ξ(r)which< 0)comparedwitharandomPoissonpoint quantify the ‘excess Figure 4 | Galaxy two-point correlation function, y(r), at the present epoch the data – there were no computers! distribution, for whichprobability’ξ(r)=0 of finding.Itisworthtomentionthatin a galaxy as a function of separation r. Red symbols (with vanishingly small Poisson Figure 5 | Galaxy clustering as a function of luminosity and colour. a, the error bars) show measurements for model galaxies brighter than two-point correlation function of our galaxy catalogue at z 0 split by It was Totsuji and Kihara (1969) and, independently, statistical mechanics the correlation function normally used is ¼ (or halo) close to another one M K 223, where M K is the magnitude in the K-band. Data for the large luminosity in the b J -band filter (symbols with 1-j error bars) Brighter Peebles (1974b) who were first to present a computer- g(r)=1+ξ(r) which is called the radial distribution functionspectroscopic¼ redshift survey 2dFGRS (ref. 28) are shown as blue diamonds galaxies are more strongly clustered, in quantitative agreement with based analysis of a complete catalog of galaxies. Tot- together with their 1-j error bars. The SDSS (ref. 34) and APM (ref. 31) observations33 (dashed lines). Splitting galaxies according to colour (b), we surveys give similar results. Both for the observational data and for the find that red galaxies are more strongly clustered with a steeper correlation suji and Kihara used the published Lick counts in cells simulated galaxies, the correlation function is very close to a power law for slope than blue galaxies. Observations35 (dashed lines) show a similar trend, r # 20h21 Mpc. By contrast, the correlation function for the dark matter although the difference in clustering amplitude is smaller than in this from Shane and Wirtanen (1967), while Peebles and cowork- (dashed green line) deviates strongly from a power law. particular semi-analytic model. 5 ers analysed a number of catalogs: the Reference catalog This property is called stationarity in point field statistics. 633 © 2005 Nature Publishing Group Springel+05